A link polynomial via a vertex-edge-face state model

TL;DR

This paper introduces a new 2-variable link polynomial combining Alexander and Jones polynomial models, with a refined version for certain 3-space links containing a Hopf link.

Contribution

It constructs a novel 2-variable polynomial linking Alexander and Jones polynomials and refines it for specific 3-space links with Hopf sublinks.

Findings

Proposes a 2-variable link polynomial $W_L$

Conjectures $W_L$ is a product of Alexander and another polynomial

Refines $W_L$ for links containing a Hopf sublink

Abstract

We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$ to an ordered set of 3-variable polynomials for those links in 3-space which contain a Hopf link as a sublink.